Quadratic constructions of copulas

In the paper we introduce and study quadratic constructions of copulas based on composition of a copula and some quadratic polynomial. We characterize all quadratic polynomials whose composition with an arbitrary copula always results in a copula. Due to this result, we can assign a two-parametric class KCc,d of copulas with parameters (c,d)(c,d) in a certain subset Ω⊆R2Ω⊆R2 to each copula C. Moreover, we also determine all copulas invariant with respect to the quadratic constructions presented. This investigation brings two interesting parametric classes of copulas. We show that the union of these two classes is equal to the so-called Plackett family of copulas. We add some properties of these copulas and their statistical consequences.